On the exact solution for the Schrödinger equation

Abstract

For almost 75 years, the general solution for the Schr\"odinger equation was assumed to be generated by a time-ordered exponential known as the Dyson series. We discuss under which conditions the unitarity of this solution is broken, and additional singular dynamics emerges. Then, we provide an alternative construction that is manifestly unitary, regardless of the choice of the Hamiltonian, and study various aspects of the implications. The new construction involves an additional self-adjoint operator that might evolve in a non-gradual way. Its corresponding dynamics for gauge theories exhibit the behavior of a collective object governed by a singular Liouville's equation that performs transitions at a measure $0$ set. Our considerations show that Schr\"odinger's and Liouville's equations are, in fact, two sides of the same coin, and together they become the unified description of quantum systems.